Classics in Applied Mathematics 53
Perturbation Bounds for Matrix Eigenvalues contains a unified exposition of spectral variation inequalities for matrices. The text provides a complete and self-contained collection of bounds for the distance between the eigenvalues of two matrices, which could be arbitrary or restricted to special classes. The books emphasis on sharp estimates, general principles, elegant methods, and powerful techniques, makes it a good reference for researchers and students.
For the SIAM Classics edition, the author has added over 60 pages of new material, which includes recent results and discusses the important advances made in the theory, results, and proof techniques of spectral variation problems in the two decades since the books original publication.
Audience
This updated edition is appropriate for use as a research reference for physicists, engineers, computer scientists, and mathematicians interested in operator theory, linear algebra, and numerical analysis. The text is also suitable for a graduate course in linear algebra or functional analysis.
About the Author
Rajendra Bhatia is a Professor at the Indian Statistical Institute and has held visiting positions at several universities around the world. He is a J. C. Bose National Fellow and a fellow of the Indian Academy of Sciences and of the Indian National Science Academy. Professor Bhatia received the Bhatnagar Prize for Science and Technology in 1995. He is a Senior Editor of Linear Algebra and Its Applications and a past editor of SIAM Journal on Matrix Analysis and Its Applications.
Table of Contents
Available June 2007 / Approx. vi + 191 pages / Softcover / ISBN 978-0-898716-31-3
Differential geometry has a long, wonderful history it has found relevance in areas ranging from machinery design of the classification of four-manifolds to the creation of theories of nature's fundamental forces to the study of DNA.
This book studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole, it mixes geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences.
Differential geometry is not just for mathematics majors, it is also for students in engineering and the sciences. Into the mix of these ideas comes the opportunity to visualize concepts through the use of computer algebra systems such as Maple. The book emphasizes that this visualization goes hand-in-hand with the understanding of the mathematics behind the computer construction. Students will not only "see" geodesics on surfaces, but they will also see the effect that an abstract result such as the Clairaut relation can have on geodesics. Furthermore, the book shows how the equations of motion of particles constrained to surfaces are actually types of geodesics. Students will also see how particles move under constraints. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract.
Contents
Preface
The Point of this Book
Projects
Prerequisites
Book Features
Elliptic Functions and Maple Note
Thanks
For Users of Previous Editions
Maple 8 to 9
Note to Students
Chapter 1. The Geometry of Curves
1.1 Introduction
1.2 Arclength Parametrization
1.3 Frenet Formulas
1.4 Non-Unit Speed Curves
1.5 Some Implications of Curvature and Torsion
1.6 Green's Theorem and the Isoperimetric Inequality
1.7 The Geometry of Curves and Maple
Chapter 2. Surfaces
2.1 Introduction
2.2 The Geometry of Surfaces
2.3 The Linear Algebra of Surfaces
2..4 Normal Curvature
2.5 Surfaces and Maple
Chapter 3. Curvatures
3.1 Introduction
3.2 Calculating Curvature
3.3 Surfaces of Revolution
3.4 A Formula for Gauss Curvature
3.5 Some Effects of Curvature(s)
3.6 Surfaces of Delaunay
3.7 Elliptic Functions, Maple and Geometry
3.8 Calculating Curvature with Maple
Chapter 4. Constant Mean Curvature Surfaces
4.1 Introduction
4.2 First Notions in Minimal Surfaces
4.3 Area Minimization
4.4 Constant Mean Curvature
4.5 Harmonic Functions
4.6 Complex Variables
4.7 Isothermal Coordinates
4.8 The Weierstrass-Enneper Representations
4.9 Maple and Minimal Surfaces
Chapter 5. Geodesics, Metrics and Isometries
5.1 Introduction
5.2 The Geodesic Equations and the Clairaut Relation
5.3 A Brief Digression on Completeness
5.4 Surfaces not in R3
5.5 Isometries and Conformal Maps
5.6 Geodesics and Maple
5.7 An Industrial Application
Chapter 6. Holonomy and the Gauss-Bonnet Theorem
6.1 Introduction
6.62 The Covariant Derivative Revisited
6..3 Parallel Vector Fields and Holonomy
6.4 Foucault's Pendulum
6.5 The Angle Excess Theorem
6.6 The Gauss-Bonnet Theorem
6.7 Applications of Gauss-Bonnet
6.8 Geodesic Polar Coordinates
6.9 Maple and Holonomy
Chapter 7. The Calculus of Variations and Geometry
7.1 The Euler-Lagrange Equations
7.2 Transversality and Natural Boundary Conditions
7.3 The Basic Examples
7.4 Higher-Order Problems
7.5 The Weierstrass E-Function
7.6 Problems with Constraints
7.7 Further Applications to Geometry and Mechanics
7.8 The Pontryagin maximum Principle
7.9 An Application to the Shape of a Balloon
7.10 The Caluclus of Variations and Maple
Chapter 8. A Glimpse at Higher Dimensions
8.1 Introduction
8.2 Manifolds
8.3 The Covariant Derivative
8.4 Christoffel Symbols
8.5 Curvatures
8.6 The Charming Doubleness
Appendix A. List of Examples
A.1 Examples in Chapter 1
A.2 Examples in Chapter 2
A.3 Examples in Chapter 3
A.4 Examples in Chapter 4
A.5 Examples in Chapter 5
A.6 Examples in Chapter 6
A.7 Examples in Chapter 7
A.8 Examples in Chapter 8
Appendix B. Hints and Solutions to Selected Problems
B.1. Chapter 1: The Geometry of Curves
B.2. Chapter 2: Surfaces
B.3. Chapter 3: Curvatures
B.4 Chapter 4: Constant Mean Curvature Surfaces
B.5 Chapter 5: Geodesics, Metrics and Isometries
B.6. Chapter 6: Holonomy and the Gauss-Bonnet Theorem
B.7. Chapter 7: The Calculus of Variations and Geometry
B.8. Chapter 8: A Glimpse of Higher Dimensions
Appendix C. Suggested Projects for Differential Geometry
ISBN:978-0-88385-748-9
Hardbound, 2007
A classic book is back in print! It can be used as a supplement in a Calculus course, or a History of Mathematics course.
The first half of Calculus Gems entitles, Brief Lives is a biographical history of mathematics from the earliest times to the late nineteenth century. The author shows that Science?and mathematics in particular?is something that people do, and not merely a mass of observed data and abstract theory. He demonstrates the profound connections that join mathematics to the history of philosophy and also to the broader intellectual and social history of Western civilization.
The second half of the book contains nuggets that Simmons has collected from number theory, geometry, science, etc., which he has used in his mathematics classes. G.H. Hardy once said, "A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." This part of the book contains a wide variety of these patterns, arranged in an order roughly corresponding to the order of the ideas in most calculus courses. Some of the sections even have a few problems.
Professor Simmons tells us in the Preface of Calculus Gems: "I hold the naive but logically impeccable view that there are only two kinds of students in our colleges and universities, those who are attracted to mathematics; and those who are not yet attracted, but might be. My intended audience embraces both types." The overall aim of the book is to answer the question, "What is mathematics for? and with its inevitable answer, To delight the mind and help us understand the world."
ISBN:978-0-88385-561-4
Series:Spectrum
Series: New Mathematical Monographs (No. 4)
Paperback (ISBN-13: 9780521712293)
Also available in Hardback | eBook format
Diophantine geometry has been studied by number theorists for thousands of years, since the time of Pythagoras, and has continued to be a rich area of ideas such as Fermat's Last Theorem, and most recently the ABC conjecture. This monograph is a bridge between the classical theory and modern approach via arithmetic geometry. The authors provide a clear path through the subject for graduate students and researchers. They have re-examined many results and much of the literature, and give a thorough account of several topics at a level not seen before in book form. The treatment is largely self-contained, with proofs given in full detail. Many results appear here for the first time. The book concludes with a comprehensive bibliography. It is destined to be a definitive reference on modern diophantine geometry, bringing a new standard of rigor and elegance to the field.
* The authors have re-examined many results and much of the literature, and give a thorough account of several topics at a level not seen before in book form
* For graduate students and researchers, and is largely self-contained: proofs are given in full detail, and many results appear here for the first time
* Destined to be a definitive reference on modern diophantine geometry, bringing a new standard of rigour and elegance to the field
Contents
I. Heights; II. Weil heights; III. Linear tori; IV. Small points; V. The unit equation; VI. Rothfs theorem; VII. The subspace theorem; VIII. Abelian varieties; IX. Neron-Tate heights; X. The Mordell-Weil theorem; XI. Faltings theorem; XII. The ABC-conjecture; XIII. Nevanlinna theory; XIV. The Vojta conjectures; Appendix A. Algebraic geometry; Appendix B. Ramification; Appendix C. Geometry of numbers; Bibliography; Glossary of notation; Index.
Reviews
'This monograph is a bridge between the classical theory and a modern approach via arithmetic geometry. The authors aim to provide a clear path through the subject for graduate students and researchers. They have re-examined many results and much of the literature, and give a thorough account of several topics at a level not seen before in book form.f L'
nseignement mathematique
eThe quality of exposition is exemplary, which is not surprising, given the brilliant expository style of the elder author.f Yuri Bilu, Mathematical Review
eBombieri and Gubler have written an excellent introduction to some exciting mathematics c written with an excellent combination of clarity and rigor, with the authors highlighting which parts can be skipped on a first reading and which parts are particularly important for later material. The book also contains a glossary of notation, a good index, and a nice bibliography collecting many of the primary sources in this field.f MAA Reviews
Series: Cambridge Introductions to Philosophy
Paperback (ISBN-13: 9780521674539)
Hardback (ISBN-13: 9780521857840)
In 1931, the young Kurt Godel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Godel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic.
* An ideal textbook for philosophy and mathematics students taking a first course in mathematical logic
* Rare coverage of a number of different proofs of the core incompleteness theorem
*Includes a companion website with exercises
Contents
Preface, 1. What Godel's Theorems say; 2. Decidability and enumerability; 3. Axiomatized formal theories; 4. Capturing numerical properties; 5. The truths of arithmetic; 6. Sufficiently strong arithmetics; 7. Interlude: taking stock; 8. Two formalized arithmetics; 9. What Q can prove; 10. First-order Peano Arithmetic; 11. Primitive recursive functions; 12. Capturing funtions; 13. Q is p.r. adequate; 14. Interlude: a very little about Principia; 15. The arithmetization of syntax; 16. PA is incomplete; 17. Godel's First Theorem; 18. Interlude: about the First Theorem; 19. Strengthening the First Theorem; 20. The Diagonalization Lemma; 21. Using the Diagonalization Lemma; 22. Second-order arithmetics; 23. Interlude: incompleteness and Isaacsonfs conjecture; 24. Godel's Second Theorem for PA; 25. The derivability conditions; 26. Deriving the derivability conditions; 27. Reflections; 28. Interlude: about the Second Theorem; 29. Recursive functions; 30. Undecidability and incompleteness; 31. Turing machines; 32. Turing machines and recursiveness; 33. Halting problems; 34. The Church-Turing Thesis; 35. Proving the Thesis? 36. Looking back.